On the Dirichlet Boundary

Controllability of the 1D-Heat Equation.

Semi-Analytical Calculations and Ill-posedness Degree.

Faker Ben Belgacem∗ Sidi Mahmoud Kaber†

December 23, 2010

Abstract

The ill-posedness degree for the controlability of the one-dimensional heat equation by aDirichlet boundary control is the purpose of this work. This problem is severely (or exponen-tially) ill-posed. We intend to shed more light on this assertion and the underlying mathematics.We start by discussing the framework liable to fit an efficient numerical implementation withoutintroducing further complications in the theoretical analysis. We expose afterward the Fouriercalculations that transform the ill-posedness issue to a one related to the linear algebra. Thisconsists in investigating the singular values of some infinite structured matrices that are ob-tained as sums of Cauchy matrices. Calling for the Gershgorin-Hadamard theorem and theCollatz-Wielandt formula, we are able to provide lower and upper bounds for the largest singu-lar value of these matrices. After checking out that they are also solutions of some symmetricLyapunov (or Sylvester) equations with a very low displacement rank, we use an estimate thatimproves former Penzl’s result to bound the ratio smaller/largest singular values of these matri-ces. Accordingly, the controllability problem is confirmed to be severely ill-posed. The boundsproved here will be supported by computations made by means of Matlab procedures. At last,the well known explicit inverse of Cauchy’s type matrices allows to provide a formal exponen-tial series representation of the Dirichlet control in a long horizon controllability. That serieshas to be checked afterward whether it is convergent or not to find out whether the desiredstate is reachable or not. Here again, some examples run within Matlab will be discussed andcommented.

keywords: Controllability, HUM control, Dirichlet control, Cauchy matrices, Loewner matri-

ces, Lyapunov equation, Penzl’s type bounds, ill-posedness degree.

1 Introduction

Let I be the segment (0, π) of the real axis. Let T be a positive real number. We set Q = I×]0, T [.

The generic point in I is denoted by x and the generic time is t. The one dimensional controllability

problem we deal with reads as follows. Let yT be a given function that represents a desired state

we hope to reach at a time T . We aim at: finding a Dirichlet data u enforced at x = π satisfying

yu(x, T ) = yT (x), ∀x ∈ I, (1)

∗Universite de Technologie de Compiegne, BP 20529, 60205 Compiegne cedex, France.†UMR 7598, C.N.R.S. & Universite Pierre et Marie Curie, B.C. 187, 4 place Jussieu, 75252 Paris cedex 05, France.

1

where the symbol yv denotes the solution of the heat equation with a Dirichlet boundary condition:

∂tyv − ∂2xxyv = 0 in Q, (2)

yv(0, ·) = 0, yv(π, ·) = v on (0, T ), (3)

yv(·, 0) = 0 on I. (4)

The fact that the initial condition is fixed to zero is not really a restriction. It is adopted only for

simplification and the linear superposition principle restore that generality.

Knowledge on the controllability of the heat equation is substantially less achieved than for the

wave equation. The difficulty arisen in the heat equation is caused by the dissipation which is

mainly irreversible in opposition to the propagation which is a reversible phenomenon. Neverthe-

less, some theoretical results are already stated. May be, the main reference on the topic together

with the popular HUM method is the book by J.-L. Lions [15]. Some others are necessary surveys

for who are involved in the subject, such as [16, 12, 33] without being exhaustive. This problem

is not exactly controllable. Some states yT are not reachable. As a matter of fact, an arbitrary

yT should be considered as non-reachable. Admittedly, the problem is ill-posed in any reasonable

‘Sobolev’ framework. In contrast, controls do exist that steer the trajectory arbitrarily close to any

given state yT after a duration T . We have therefore approximate controllability (see [15, 24, 33]).

Notice that whenever, yT comes from a solution of the heat equation, the controllability is equiv-

alent to the null-controllability that is when the initial condition is given, yv(·, 0) = y0 and the

final state is yT = 0. This last pertains, in the practice, to the stabilization of dynamical systems.

Results on the null-controllability are stated by H. O. Fattorini and D. Russel (see [6]) and also by

G. Lebeau and L. Robbiano (see [13]), in one dimension for the firsts and in the general dimension

for the seconds. The major result established in there is the observability estimate that yields the

well-posedness of the null-controllability problem. The numerical counterpart of this problem has

been considered by S. Micu and E. Zuazua (see [19]) where they show the severe ill-posedness of the

problems they have to cope with computationally. Some issues treated in [19] are similar to those

we are involved in here. The methodology is however very different. Their approach, mainly based

on the analysis of some Bi-Orthogonal sequences, is the one proposed earlier in [6] and surveyed

in [24]. Here, as will be detailed later on, we rather take profit of the spectral estimates for some

known structured matrices such as Cauchy and Lowner type matrices to establish our results.

As indicated and discussed in [2] for the optimal control problem, the point is to consider a

subspace of admissible Dirichlet controls that brings facilities in the computations. That should be

a subspace of L2(0, T ) (1) that allows to define the final observation yv(·, T ) ∈ L2(I). It can not

coincide with the full Lebesgue space L2(0, T ), since the final observation yv(·, T ) may not belong

to L2(I) for some Dirichlet data in L2(0, T ) (see [15, Chap. 9]). We describe the framework we use

that turns out to be advantageous when a regularization strategy is aimed which is mandatory to

handle safely the computations. Next, comes the main concern related to the incorrect-posedness

of the controllability problem by Dirichlet controls. To state its severe ill-posedness according to

1L2(0, T ) is the Lebesgue space of square integrable functions.

2

the classification provided in [32], we perform here semi-analytical computations that transfer the

difficulty to the study of the spectrum of some Cauchy type matrices. Connecting them to Lya-

punov equations, and using the important results on the spectrum of their solution, we are able

to obtain the desired ill-posedness results. A predicted effect of it is that when a reachable yT

is perturbed by noise, the control we compute may be drastically changed, it may even blow up

exponentially fast. Additionally, owing to the explicit inverse of Cauchy matrices we are enabled

to provide the exact formal expression of the control in the case of infinite horizon controllability

which demonstrates, if needed, the exponential instability of the controllability problem. Notice

that the algebraic tools on which the analysis conducted here relies, are mainly developed by the

control community.

An outline of the paper is as follows. We describe in Section 2 the variational formulation of

the heat equation (2)-(4) derived by duality. This approach enables to account for the lack of

smoothness of Dirichlet’s data (see [12]). As a consequence, we obtain a mathematical setting to

the controllability problem in the natural space H−1(I) (2). Engineers prefer the L2(I)-setting.

Despite the practicability and effectiveness of that framework from the numerical implementation

view, particular caution should be paid to the analysis. In Section 3, we provide and discuss the

suitable tools to successfully deal with that issue. We put then the controllability problem under

an abstract form. The HUM method fails and the problem is ill-posed. This is the main subject

of Section 4. Using Fourier calculations, we check out how the controllability problem may be put

under an integral equation defined by a kernel operator. The ill-posedness degree of this equation

is therefore tightly connected to asymptotics of the singular values of the related integral operator.

This enables us to construct an infinite countable symmetric and positive definite matrix of a

Cauchy type whose eigenvalues are exactly the squares of the singular values of the operator under

investigation. Their distribution along the positive semi-axis is obtained after writing this matrix

as the solution of a Lyapunov equation with a low rank displacement. Then, using estimates

established by many authors (see [25, 9] and wide bibliographies therein) produces the desired

information on the behavior on the singular values. Their asymptotics establish the severe ill-

posedness of the controllability problem. Some Matlab computations allow to check numerically

these theoretical predictions. Owing to the inversion formula of Cauchy matrices, we provide in

Section 5 an infinite sum representation of the Dirichlet control that solves the controllability

problem for a given target yT and we check why it mostly blows up exponentially fast. Lastly,

Section 6 is dedicated to the discussion of some numerical examples to check out the theoretical

predictions.

Notation — Let X be a Banach space endowed with its norm ‖ · ‖X . We denote by L2(0, T ;X)

the space of measurable functions v from (0, T ) in X such that

‖v‖L2(0,T ;X) =(∫ T

0‖v(·, s)‖2X ds

)1/2< +∞. (5)

2The dual space of H10 (I), the Sobolev space that contains all functions that belongs to L2(I) so as their derivatives

and vanish at x = 0 and π.

3

For any positive integerm, we introduce the spaceHm(0, T ;X) of functions in L2(0, T ;X) such that

all their time derivatives up to the order m belong to L2(0, T ;X). We also use the space C (0, T ;X)

of continuous functions v from [0, T ] in X. Finally, we consider the Sobolev space H1(I) of all the

functions that belong to L2(I) together with their first derivatives. The space H10 (I) defined in

footnote (2), is then the closure in H1(I) of the space D(I) of infinitely differentiable functions

with a compact support in I, and by H−1(I) its dual space.

2 Controllability in H−1(I)

Let us first recall the variational formulation of the parabolic system (2)-(4). As indicated in the

introduction, our willing is to consider controls in the Lebesgue space L2(0, T ). Because of the lack

of regularity on the Dirichlet datum v, the variational form is proceeded by a duality argument.

We follow the transposition method detailed in [15, Th. 9.1]. Let then f be arbitrarily given in

L2(Q). Consider rf as the solution of the backward heat equation

−∂trf − ∂2xxrf = f in Q,

rf (0, ·) = 0, rf (π, ·) = 0 on ∂(0, T ),

rf (·, T ) = 0 in I.

The solution rf does exist and is unique in L2(0, T ;H10 (I)) ∩ C (0, T ;L2(I)) (see [17, Chap. 4, Th.

1.1]). Provided that the boundary is regular enough which is assumed from now on, the normal

derivative ∂xrf (π, ·) belongs to L2(0, T ) and is subjected to the stability

‖∂xrf (π, ·)‖L2(0,T ) ≤ C ‖f‖L2(Q). (6)

Applying the transposition method to problem (2)-(4), we come up with the following variational

equation: find yv in L2(Q) such that

∫

Qyv(x, t)f(x, t) dx dt = −

∫

(0,T )v(t)

(∂xrf

)(π, t) dt, ∀f ∈ L2(Q). (7)

The equivalence between (7) and problem (2)-(4) has been checked for instance in [2, Proposition

2.2]. The existence and uniqueness for yv are direct consequences of (6) and Riesz’ Theorem (see

[12, 14]). Then, for any v in L2(0, T ), problem (7) has a unique solution yv in L2(Q) which belongs

to C ([0, T ];H−1(I)).

Now, we are in position to rewrite problem (1) in a rigorous mathematical words before adding

some modifications to the functional setting for practical reasons that will be commented later on.

The controllability problem consists then in: finding u ∈ L2(0, T ) such that

yu(·, T ) = yT (·), in H−1(I). (8)

When the data yT suffers from some perturbations, existence for problem (8) may fail. Noisy data

are unavoidable, this may be caused by erroneous measurement or preliminary processes to prepare

4

the data before feeding any computational program. More, even when yT is an ideal mathematical

fixed state, when switching to computation, the numerical treatment of that yT generates with

certainty some disturbances. The consequence is that in simulations users can not spare using a

regularization strategy necessary for obtaining pertinent results. (see [29, 5]). In crude computa-

tions, the noise generates oscillations on the approximated control. Their intensity depends on the

bad conditioning of the discrete controllability problem. The indicator to quantify how it is badly

conditioned is the ill-posedness degree that has been introduced by G. Wahba (see [32]). This is the

main subject here. We intend to demonstrate the severe ill-posedness of the controllability problem.

The choice of the framework where to proceed requires some comments at this stage. Most

users interested in prefer to consider an L2(I) setting rather than H−1(I) which seems to be the

natural space. Why? The answer is strongly related to the numerical implementation and then to

practical engineering issues. As a matter of fact, computing the L2-norm of (yv(·, T )), necessarywhen regularization is required, is an easy matter while the evaluation of H−1-norm arises some

difficulties practitioners prefer to avoid. Moreover, a comprehensive analysis of the regularization

strategy may also be achieved in the modified framework. This is the content of the forthcoming

work together with the study of the effect of the penalization of the Dirichlet control by a Fourier

boundary condition.

3 Controllability in L2(I)

The key of the analysis in the L2(I)–framework is a Green formula written for non-regular func-

tions. It has been stated in [2]. Nevertheless, to be self contained and in order to introduce some

technical tools related to, we need to provide that result.

Let yv be the solution of problem (7). We have already mentioned that yv(·, T ) fails to belong

to L2(I) for some v. We define then the unbounded operator B in L2(0, T ),

Bv = yv(·, T ).

The domain D(B) is hence composed of functions v ∈ L2(0, T ) for which yv(·, T ) lies in L2(I),

D(B) ={v ∈ L2(0, T ); yv(·, T ) ∈ L2(I)

}.

Following [2, Lemma 2.4] it can be checked that B is closed.

Lemma 3.1 The operator B is closed. As a consequence, D(B) is dense in L2(0, T ) and the graph

norm

‖v‖D(B) =(‖v‖2L2(0,T ) + ‖yv(·, T )‖2L2(I)

)1/2,

determines a Hilbert structure on D(B).

Proof: Let (vm, Bvm)m be a convergent sequence in L2(0, T ) × L2(I) and (v, ψ) be the limit.

That yvm belongs to C (0, T ;H−1(I)) implies that yvm(·, T ) converges towards yv(·, T ) in H−1(I).

5

Now, since L2(I) is continuously embedded in H−1(I) we derive that ψ = yv(·, T ). Hence, not onluyv(·, T ) ∈ L2(I) but also Bv = ψ. The proof is complete.

A further result is that the adjoint operator B∗ is well defined with a dense domain D(B∗) in

L2(I). It is specified by the identity

(Bv, ψ)L2(I) = (v,B∗ψ)L2(0,T ), ∀(v, ψ) ∈ D(B)× D(B∗). (9)

A closed form of it may be provided (see for instance [12, 2]). Let ψ be given in L2(I) and denote

by qψ the unique solution of the problem

−∂tqψ − ∂2xxqψ = 0 in Q, (10)

qψ(0, ·), qψ(π, ·) = 0 on (0, T ), (11)

qψ(·, T ) = ψ on I. (12)

The function qψ does exit in L2(0, T ;H1(I)) ∩ C (0, T ;L2(I)) (see [17, Chap. 4]). The following

result is stated for instance in [2, Lemma 2.5]. Nevertheless, we choose to check it again to be

self-contained. Besides, the proof is really short.

Lemma 3.2 The domain of B∗ is given by

D(B∗) ={ψ ∈ L2(I); ∂xqψ(π, ·) ∈ L2(0, T )

},

and B∗ is defined as follows

B∗ψ = −∂xqψ(π, ·).

Proof: For a given function v in D(0, T ), multiply (10) by yv and integrate by part on Q. This is

legitimate since we dispose of enough smoothness. We derive that

(Bv, ψ)L2(I) =

∫

Iyv(x, T )ψ dx = 〈v, (−∂xqψ(π, ·))〉.

Using (9) and a density argument yields that B∗ψ = −∂xqψ(π, ·) in D ′(0, T ). In view of the fact

that B∗ψ lies in L2(0, T ), we deduce that ∂xqψ(π, ·) belongs also to L2(0, T ). This concludes the

proof.

Remark 3.1 The space D(B∗) is a Hilbert space when endowed with the graph norm

‖ψ‖D(B∗) =(‖ψ‖2L2(I) + ‖∂xqψ(π, ·)‖2L2(0,T )

)1/2, ∀ψ ∈ D(B∗).

Given that B∗ is injective, the mapping ψ 7→ ∂xqψ(π, ·) is a norm in D(B∗). The Hilbert space Hobtained by the completion of D(B∗) with respect to this norm can not contain L2(I). Otherwise,

B∗ would be bounded. However, as already noticed in [19], the regularity of (∂xqψ(π, ·)) is not

affected by the (non-)smoothness of ψ away from x = π. As a result, H may contain non-regular

final states ψ provided that their singularities or high oscillations are located in [0, π[.

6

Remark 3.2 The invertibility of B∗ pertains to the Cauchy problem for the backward heat equation

where the final condition is to be reconstructed from over-specified Cauchy boundary conditions at

x = π. This is known in the literature as the sideways problem. That problem has at most one

solution by the unique continuation theorem [26]. This means that B∗ is injective and therefore

Ker B∗ = {0}. (13)

Homogeneous Cauchy data on {π} × (0, T ) for the problem satisfied by qψ yields that qψ = 0 and

thus ψ = 0. As a consequence, the range R(B) is dense in L2(I). The approximate controllability

holds (see [12, 33]). Moreover, it will be checked that it has a non-closed range and can not be

continuously invertible.

A Green formula is the by-product of the construction of B and B∗. We aim a general version

for it. We then assume given ψ in D(B∗) and f in L2(Q). We introduce the solution q(ψ,f) in

L2(0, T ;H10 (I)) ∩ C (0, T ;L2(I)) of the problem

−∂tq(ψ,f) − ∂xxq(ψ,f) = f in Q,

q(ψ,f)(0, ·) = 0, q(ψ,f)(π, ·) = 0 on (0, T ),

q(ψ,f)(·, T ) = ψ on I.

We have the following result

Lemma 3.3 Let f be in L2(Q), v in D(B) and ψ in D(B∗). The following Green formula holds

∫

Qf(x, t)yv(x, t) dxdt+

∫

Iψ(x)yv(x, T ) dx+

∫

(0,T )v(t) ∂xq(ψ,f)(π, t) dt = 0.

Proof: It may be found in [2, Proposition 2.5 and Corollary 2.6].

That the operator B is defined in the L2-framework, the controllability problem (8) can be

reworded as an equation involving B. Given a data yT ∈ L2(I). The problem consists in: finding

u ∈ D(B) such that

Bu = yT , in L2(I). (14)

This is the very problem we deal with in the subsequent, unless explicitly contradicted. Whenever

we talk about the controllability problem we mean this one and not (8) (in H−1(I)).

It is worth to notice that the range subspace R(B) is dense in L2(I) as made Remark 3.2.

However, it will be more instructive to know that the complementary of R(B), composed of the

non-reachable states yT , is also dense in L2(I). This means that any perturbation of any reachable

state yT = yv(·, T ), for some v, may produce a non-reachable perturbed final state yT 6∈ R(B).

This result is important for the analysis of regularization strategies which are mandatory for the

computational treatment of the problem given its severe ill-posedness.

Lemma 3.4 The set of non-reachable states L2(I) \ R(B) is dense in L2(I).

7

Proof: Let us proceed by contradiction. Assume then that L2(I) \ R(B) is not dense in L2(I).

Hence, R(B) has a non-empty interior. As it is a vector subspace, there is no other possibility

than it coincides with the entire space L2(I). The operator B will be therefore surjective. Now,

suppose that D(B) is endowed with the graph norm ‖ · ‖D(B) provided in Lemma 3.1, it is a Hilbert

space. Consider hence B as an operator mapping D(B) in L2(I). It is obviously injective and

continuous. In addition, it is surjective and then bijective. Owing to the open map theorem it

defines an isomorphism. In particular, we obtain that

‖v‖L2(0,T ) ≤ C‖Bv‖L2(I), ∀v ∈ D(B).

As a result, B−1 will be a bounded operator from L2(I) into L2(0, T ) which is necessarily false.

L2(I) \ R(B) is then dense in L2(I). The proof is complete.

Remark 3.3 As an alternative to (14), duality for the convex optimization suggests to look at the

minimization problem

minχ∈D(B∗)

J∗(χ) = minχ∈D(B∗)

1

2‖∂xqχ‖2L2(0,T ) − (yT , χ)L2(I).

Whenever the minimum is reached, let us say for ψ ∈ D(B∗), using Green’s formula of Lemma

3.3 we state that the control u† = B∗ψ = −∂xqψ. It is the HUM control of problem (14). As

well known, and will be checked again below, the controllability problem has an infinite number of

solutions. The solution u†, issued from the duality is the one, among all the possible solutions, that

has the minimum norm in L2(0, T ).

Remark 3.4 Users may be tempted to apply HUM theory to the controllability problem (8), set

in H−1(I). Standard Green’s formula (3) results in the reduced problem that reads as follows: find

u ∈ L2(0, T ) such that

∫

(0,T )u(t) ∂xqψ(π, t) dt = 〈yT , ψ〉H−1,H1

0, ∀ψ ∈ H1

0 (I).

The symbol qψ is for the solution of (10)-(12). The well posedness requires the observability estimate

that is

‖ψ‖H1(I) ≤ C‖∂xqψ(π, ·)‖L2(0,T ), ∀ψ ∈ H10 (I).

Unfortunately, this does not hold. Switching to the space L2(I) in (8), instead of H−1(I), does not

change that much the situation. Indeed, although we need some weaker observability inequality

‖ψ‖L2(I) ≤ C‖∂xqψ(π, ·)‖L2(0,T ), ∀ψ ∈ D(B∗),

this is false too. The best inequality of similar kind we know of so far is a Carleman estimate

established by M. Klibanov in [11, 2006].

3Green’s formula in Lemma 3.3 serves when the controllability equation is written in L2(I) and the control issought for in D(B).

8

4 The Severe Ill-posedness

Needless to say that the ill-posedness of problem (14) is governed by the properties of the operator

B. On account of its unboundedness, the degree of ill-posedness of the controllability problem, in

the classification of G. Wahba (see [32]), is inferred from the speed at which the ratio (larger/lower)

singular values of B blows up to infinity. We mainly pursue the behavior of the singular values

of B. The chief tool of our study is the Fourier analysis and the spectra of some structured matrices.

4.1 B and B∗ as Kernel Operators

Since B and B∗ have the same non-vanishing singular values we choose to focus rather on B∗ and

then infer the expression of B by duality. After all, the equation defined by B∗ is interesting in

itself since it is the mathematical modeling of the inverse problem, well known as the sideways

equation, that consists in reconstructing the initial data from the knowledge of the Cauchy bound-

ary conditions (see [1]).

Notice that the construction of B∗ is based on the backward heat equation. For easiness, especially

when the Laplace transform is used, we prefer to handle the direct heat equation. All the compu-

tations and constructions are also valid modulo some slight modifications for the backward heat

problem. In particular the structured matrices we have to deal with are not affected by progressive

or backward problems. We consider then the problem

∂tq − ∂2xxq = 0 in I × (0, T ), (15)

q(0, ·) = 0, q(π, ·) = 0 on (0, T ), (16)

q(·, 0) = ψ on I. (17)

The ‘adjoint’ operator B∗ is still expressed as follows

B∗ψ = −∂xq(π, t), in (0, T ).

Remark 4.1 In this context, the controllability problem related to (15)-(17) is therefore governed

by the backward heat equation as that state equation

−∂tyv − ∂2xxyv = 0 in Q,

yv(0, ·) = 0, yv(π, ·) = v on (0, T ),

yv(·, T ) = 0 on I.

The control is the Dirichlet boundary condition at point π, still denoted u, that realizes y(·, 0) = y0,

where y0 ∈ L2(I) is the desired state. Of course, the overall results we state hereafter are valid as

well for problem (14).

The methodology followed here has been used earlier. We refer for instance to [6, 22, 19, 33].

Given that (sin(kx))k≥1 is an orthogonal basis in L2(I), we may expand ψ, as follows

ψ(x) =∑

k≥1

ψk sin(kx) in I. (18)

9

Plug it into problem (15)-(17) and all computations conducted, B∗ may put under a kernel operator

form

(B∗ψ)(t) =∑

k≥1

(−1)k+1ke−k2tψk =

∫

IK(t, x)ψ(x) dx.

The kernel K(·, ·) is defined to be

K(t, x) =2

π

∑

k≥1

(−1)k+1ke−k2t sin(kx), in I × (0, T ).

It can be checked directly that

D(B∗) ={ψ ∈ L2(I);

∑

k≥1

∑

m≥1

(−1)k+mkm

k2 +m2(1− e−(k2+m2)T )ψkψm <∞

},

and that Ker B∗ = {0}. Things are a little bit more subtle when we are involved in the range

R(B∗) which is an important issue upon which depends the (non)uniqueness for the controllability

problem (14). In fact, it allows for the determination of Ker B. In view of the expression of

B∗ψ the sequence of functions (e−k2t)k≥1 is total in R(B∗). Since the (k−2)k≥1 is summable, then

according to a variant of the Muntz theorem (see [28]) it comes out that (e−k2t)k≥1 can not be

dense in L2(0, T ), neither is R(B∗) in L2(0, T ).

Remark 4.2 The same Muntz theorem enables one to check that the co-dimension of R(B∗) in

L2(0, T ) is infinite. Actually, a more accurate theorem by Borwein-Erdelyi in [3, Theorem 6.4]

implies that any function v in the closure R(B∗) is analytic in ]0, T ] and admits the following

representation

v(t) =∑

k≥1

vke−k2t, ∀t ∈]0, T ].

That v belongs to L2(0, T ) yields the condition

∑

k≥1

∑

m≥1

1− e−(k2+m2)T

k2 +m2vkvm <∞.

Be aware that these results are concerned with the controllability of the backward heat equation in

Remark 4.1. When the progressive equation is considered u† will be analytic on [0, T [ except probably

at t = T . The important observation to retain here is that any reachable state yT may be realized by

an analytic control on [0, T [. Recall that the reachable states yT are all indefinitely smooth in [0, π[.

They may contain a singularity at x = π. That singularity contaminates the ‘minimal’ control only

at the final time t = T . At last, notice that this holds true for any initial condition y0 ∈ L2(I)

and not only when y = 0. A suitable superposition principle applied to the problem allows such a

generalization.

Now, let v ∈ D(B) be given and denote by v(T ) its trivial extension to the semi-axis R+. The

operator B can also be expressed as a kernel operator that is

(Bv)(x) =

∫

R+

K(t, x)v(T )(t) dt =2

π

∑

k≥1

(−1)k+1[kv(T )(k

2)]sin(kx), in I. (19)

10

The symbol v(T ) is used for the Laplace transform of the extended function v(T ) ∈ L2(R+). We

deduce immediately an explicit characterization of D(B) that is

D(B) ={v ∈ L2(0, T );

∑

k≥1

[kv(T )(k

2)]2<∞

}.

Moreover the kernel of B is specified as follows

Ker B ={v ∈ D(B); v(T )(k

2) = 0, ∀k ≥ 1}.

Remark 4.3 To have a deeper insight about what really happens, let us consider the particular

case T = ∞. One can construct a collection of functions (va)a>1 that belong to Ker B. Indeed, it

can be checked that the function

va(t) =√π t−3/2 exp(−(a− 1)π2

4t)[sin(

π2√a

2t) +

√a cos(

π2√a

2t)]

lies in L2(R+) and its Laplace transform coincides with (see [21, example 7.13, p. 279])

va(p) = 2 exp(−π√ap) sin(π√p)

Then we have that va ∈ Ker B for any a > 1 and the dimension of Ker B is thus infinity.

4.2 Infinite Matrices of BB∗ and BB

∗

Now, the computation of the singular values of B∗ requires the specification of the kernel function,

we denote by G(·, ·), of the operator BB∗. Recall that BB∗ is the HUM operator. The HUM

control, whenever its existence is ensured, is given by u† = B∗(BB∗)−1yT , for a reachable state yT .

A closed form of the kernel G(·, ·) is provided by

G(x, y) =

∫

(0,T )K(t, x)K(t, y) dt

=4

π2

∑

k≥1

∑

m≥1

(−1)k+mkm

k2 +m2(1− e−(k2+m2)T ) sin(kx) sin(my), x, y ∈ I.

Let then ψ be given in D(BB∗). Using the expansion in (18) there holds that

(BB∗ψ)(x) =

∫

IG(x, y)ψ(y) dy

=2

π

∑

k≥1

∑

m≥1

(−1)k+mkm

k2 +m2(1− e−(k2+m2)T )ψm sin(kx), ∀x ∈ I.

The eigenvalues of the operator BB∗ are exactly those of the infinite-dimensional matrix B∞ =

(bkm)k,m≥1 where the entries a provided by

bkm =2

π(−1)k+m

km

k2 +m2(1− e−(k2+m2)T ), ∀k,m ≥ 1. (20)

The matrix B∞ may be viewed as an unbounded operator in ℓ2(R)(4) which is a representation of

BB∗ on the Fourier basis. It is symmetric and non-negative definite.

4ℓ2(R) is the space the numerical sequences that are square summable.

11

Remark 4.4 Conducting similar computations for the operator B∗B is of course possible. It may

be defined by the kernel operator

H(t, τ) =∑

k≥1

k2e−k2(t+τ), ∀t, τ ∈ (0, T ).

In addition, the eigenvectors should be sought for in R(B∗) and yield to the spectral decomposition

of the matrix (B∞)′ = (b′km)k,m≥1 where

b′km =2

π

k2

k2 +m2(1− e(k

2+m2)T ), ∀k,m ≥ 1.

It can be checked that B∞ and (B∞)′ are equivalent infinite-dimensional matrices. Indeed, there

hold that

(B∞)′ = QB∞Q−1.

The matrix Q is diagonal with qkk = (−1)kk. Of course, a particular caution should be paid to the

domain in ℓ2(R) of each of them.

4.3 Asymptotics of Singular Values. Severe Ill-posedness

The subsequent is to exhibit asymptotics of the eigenvalues of the infinite-dimensional Gram matrix

B∞ = (bkm)k,m≥1. From now to the end the index ∞ will be dropped off which does not modify at

all the results we have in mind. The entries of B are re-transcribed as follows

bkm = (−1)k+mkm

k2 +m2(1− e−(k2+m2)T ), ∀k,m ≥ 1.

The constant term 2/π involved in the expression of bkm given in (20) is removed. This has no real

incidence on the analysis we undertake here. Define the new matrix Z = (zkm)k,m≥1 where

zkm =km

k2 +m2(1− e−(k2+m2)T ), ∀k,m ≥ 1.

Obviously B and Z are equivalent and have the same spectrum. Studying Z (actually the truncated

form of it) is advantageous because it has positive entries and the Perron-Frobenius theory allows

to obtain further informations about the largest eigenvalue (of the truncated Z introduced here

below) called the Perron-Frobenius root (see [30]). The matrix B and consequently Z inherit the

properties of the operator BB∗ and are therefore both non-negative definite. As a result, their

eigenvalues which are common are all non-negative. The matrix Z may be written as the difference

of two Cauchy matrices. The entries of those two Cauchy matrices are respectively

(km

k2 +m2

)

k,m≥1

,

((ke−k

2T )(me−m2T )

k2 +m2

)

k,m≥1

.

Next, let us introduce the infinite dimensional diagonal matrix D = diag {k2}k≥1 and vectors

ℓ = (k)k≥1 and ℓ′ = (ke−k2T )k≥1. Set L = ℓℓ∗ − (ℓ′)(ℓ′)∗. It is readily checked that the matrix Z

satisfies the Lyapunov equation

DZ + ZD = L. (21)

12

Now, to find out what happens for the eigenvalues of Z, we focus on the truncated matrix ZN =

(zkm)1≤k,m≤N . It is the solution of the finite dimensional Lyapunov equation that is

DNZN + ZNDN = LN . (22)

All vectors and matrices are here truncated in an obvious way. Matrix equations (22) and (21) are

also called symmetric Sylvester equations. They currently arise in the controllability/observability

theory of Linear Time-Invariant systems.

The asymptotic expansion of the eigenvalues of Z are strongly connected to the properties of

the diagonal matrix DN and to the rank p of LN , called the displacement rank of equation (22).

Solving it pertains to the third Zolotarev problem. Bounds will be derived after applying results

assembled, sorted, commented and improved in two interesting Ph.D. dissertations, J. Sabino [25,

2006] and A. Gryson [9, 2009]. The authors propose also a wide bibliography on the subject. Let us

first denote by ((µN )k)1≤k≤N the eigenvalues and ((σN )k)1≤k≤N the singular values of ZN ordered

decreasingly. The following result holds

Proposition 4.1 We have that

√logN

2≤ (σN )1 ≤

√N

2. (23)

The singular values ((σN )k)1≤k≤N satisfy the bound

(σN )2k+1 ≤ C√N exp

(− π2k

4 log(2N)

), 1 ≤ k ≤ [(N − 1)/2]. (24)

Proof: The bound on the Perron-Frobenius root of ZN which is (µN )1 may be established owing

to the double formula (see [18, Chapter 8]),

min1≤k≤N

∑

1≤m≤N

km

k2 +m2(1− e−(k2+m2)T ) ≤ (µN )1 ≤ max

1≤k≤N

∑

1≤m≤N

km

k2 +m2(1− e−(k2+m2)T ).

The right bound comes from the Gershgorin-Hadamard circle theorem while the left one is directly

issued from Collatz-Wielandt formula. We deduce that

logN

2≤ (µN )1 ≤

N

2. (25)

The bound (23) is established. The first thing we learn from these bounds is that the large

eigenvalues blows up with N . This is due to (and expresses) the unboundedness of the operator Z.

Before stating (24), observe first that the displacement rank of equation (22) is two, p = 2. Calling

for Theorem 2.1.1 of [25, pp 39-40] (see also [25, Formula (2.14) in p. 46]) yields the following

bound(µN )pk+1

(µN )1=

(µN )2k+1

(µN )1≤ C exp

(− π2k

log(4κ(DN ))

). (26)

The constant C is independent of N . The symbol κ(DN ) is for the condition number of the matrix

DN . This bound valid for general Lyapunov equations is related to the matrix DN while the

13

eigenvalue (µN )1 is connected to the right hand side of the equation. Notice how the displacement

rank influences the counting of the eigenvalues.

Given that the condition number κ(DN ) is easily computed for diagonal matrices, it coincides with

the ratio of the maximal and the minimal diagonal terms and equals then to N2. Hence we obtain

that

(µN )2k+1 ≤ C(µN )1 exp

(− π2k

2 log(2N)

).

Using the right bound in (25) we obtain that

(µN )2k+1 ≤ CN exp

(− π2k

2 log(2N)

).

The proof is complete after recalling that (µN )k = [(σN )k]2.

Remark 4.5 When j = 2k + 1 ≈ θN with 0 < θ ≤ 1 we have that

(σN )j ≤ C√N exp

(− π2θN

8 log(2N)

)

These singular values decrease toward zero exponentially fast. To summarize, the largest singular

value grows up with N , at least like a constant times√

log(N)(5), while the smallest eigenvalues

decay exponentially fast. The bound in (26) yields that the ratio between the extreme singular values

(σN )1 and (σN )N is bounded from below,

(σN )1(σN )N

≥ C exp

(π2N

8 log(2N)

)

This lower bound confirms the severe ill-posedness of the controllability problem.

Remark 4.6 We emphasize here on the fact that the controllability problem is intrinsically severely

ill-posed, no matter the support of the control coincides with the whole boundary of the domain or

not. This is in contrast with the distributed control (see [20]), where the problem becomes mildly

ill-posed if the control is operating on the whole domain.

Remark 4.7 Users may be interested in the behavior of the singular values ((σN )k)1≤k≤N with

respect to the final time T . It may hence be indicated that the constant C in (24) is uniformly

bounded with respect to T . However, in case where a sharp study is intended, the dependence of

the bounds on ((σN )k)1≤k≤N may be inferred from formula (26) where (σN )1 should be accurately

estimated with respect to T . A simple look at the Gershgorin-Hadamard bound on (µN )1 shows

that it decays toward zero when T comes close to zero. This means that the exact singular values

(σk)k∈Z is shifted toward zero for small T . This fact is expected as it indicates the higher difficulty

to realize the desired state yT in a shorter time T .

All the ingredients required to provide the main result involved with the ill-posedness degree of

the controllability of the one dimensional equation. We have the following

Theorem 4.2 The controllability problem (14) is severely ill posed.

5It will be checked out numerically that it actually grows like√N .

14

5 Solution of the Controllability Problem

The aim we are assigned here is to solve the controllability problem (14) in the context of Fourier

computations. Here we need to express the operator B in respect with the progressive heat equation.

Let then v be in L2(0, T ), we set v(·) = v(T − ·). It comes out that

(Bv)(x) =2

π

∑

k≥1

(−1)k+1[kv(T )(k

2)]sin(kx), in I.

We look for a Dirichlet control u that drives the solution yu from 0 at time t = 0 to a fixed state

yT ∈ L2(I) at time t = T . Let us decompose the data yT ∈ L2(I) as follows

yT (x) =∑

k≥1

(yT )k sin(kx) in I.

On account of the expression (19), we obtain the infinite family of equations on u that is

u(T )(k2) = (−1)k+1π

2

(yT )kk

, ∀k ≥ 1. (27)

Recall that, in general, we look for the control u† that belongs to R(B∗). Owing to the represen-

tation (4.2) of the functions in R(B∗), we consider the following expansion

u†(t) =∑

m≥1

(u†)m e−m2(T−t), ∀t ∈ (0, T ).

Here, it was necessary to change t into (T − t) to take into account that we are controlling the

progressive equation. Plugging it into equation (27) yields that

∑

m≥1

1− e−(k2+m2)T

k2 +m2(u†)m = (−1)k+1π

2

(yT )kk

, ∀k ≥ 1. (28)

The problem in the form (27) looks like the inversion of some discrete Laplace Transform or a

moment problem where the solution is compactly supported. It may be formally solved by means

of the biorthogonal basis related to (e−k2t)k≥1 (see [24]). The point is that the formal expansion we

obtain for the solution u† will blow up exponentially fast. Here, in order to have a better insight

about what really happens we conduct explicit computations and use the inversion of some Cauchy

type matrices. This turn out to be possible when the controllability is aimed at a sufficiently long

horizon. We mean that T is large enough to justify the neglecting of the exponential terms in the

coefficients of (28). We are then left with the equation to solve which is somehow simpler

∑

m≥1

1

k2 +m2(u†)m = (−1)k+1π

2

(yT )kk

, ∀k ≥ 1. (29)

The methodology we follow consists first in realizing a truncation to a large cut-off N , solve the

finite dimensional square system and finally to pass formally to the limit. As things stand, the

finite dimensional system to tackle reads as

∑

1≤m≤N

1

k2 +m2(u†)m = (−1)k+1π

2

(yT )kk

, ∀1 ≤ k ≤ N. (30)

15

The symmetric matrix involved in this system, denoted by CN = (ckm = 1/(k2 +m2))1≤k,m≤N , is

a Cauchy matrix. The inverse (CN )−1 may be achieved explicitly. Indeed, we have that (see [27])

(CN )−1 = DNCNDN .

The matrix DN = diag (dk)1≤k≤N is diagonal and the entries are given by

dk =k2∏Nm=1(1 +

k2

m2)

∏Nm=1,m 6=k(1−

k2

m2)

, ∀k(1 ≤ k ≤ N).

Now, for a large cut-off N ≈ ∞, we switch to infinite products. All calculations achieved(6), we

obtain the simplified closed form of those diagonal cœfficients,

dk =2

π(−1)kk sinh(kπ), ∀k(1 ≤ k ≤ N).

Returning to the inverse matrix CN = (ckm)1≤k,m≤N , we obtain that

ckm = dkckmdm =4

π2(−1)k+m

km

k2 +m2sinh(kπ) sinh(mπ), k,m ≥ 1.

All the ingredient are thus available to derive the coefficients (u†)k of the control u†

(u†)k =∑

m≥1

(−1)m+1

mckm(yT )m =

4

π2(−1)k+1k sinh(kπ)

∑

m≥1

sinh(mπ)

k2 +m2(yT )m.

For u† to lie in L2(I) , one has to check the bound given in the end of Remark 4.1, skipped over be-

cause the expression is pretty long. We limit ourselves to the following observation, that condition

requires a high regularity on that data yT . Intuitively, the subspace of reachable states appears

quiete shrunk (although dense) in L2(I). Moreover, using this formula, we see that a small yT ,

assimilated to a noise, may produce a solution u† that blows up exponentially fast.

Remark 5.1 In a short horizon controllability, we observe that the matrix related to the truncated

system of (28) whose entries are

(1− e−(k2+m2)T

k2 +m2

)

1≤k,m≤N

may be decomposed as the product of two matrices defined by

Diag (e−k2T )1≤k≤N ,

(ek

2T − e−m2T

k2 +m2

)

1≤k,m≤N

.

6We use the well known infinite product formulae

sin(πζ)

πζ=

∞∏

m=1

(1− ζ2

m2), ∀ζ ∈ R,

sinh(πζ)

πζ=

∞∏

m=1

(1 +ζ2

m2), ∀ζ ∈ R.

16

The first is diagonal and then easy to invert while the second is again a particular structured

matrix of Loewner type. Different processes do exist to compute its inverse either analytically or

numerically using some particular algorithms. Given that it would be long to expose either of both

methods we simply refer to [31].

Remark 5.2 Realizing similar calculations for the null-controllability yields a problem that looks

like equation (28). Recall that for the null-controllability the initial data is fixed to y0 while the

final target is yT = 0. The linear problem to solve on u† may be written as

∑

m≥1

1− e−(k2+m2)T

k2 +m2(u†)m = (−1)k+1e−k

2T (y0)kk

, ∀k ≥ 1.

It may be put under the following form

∑

m≥1

ek2T − e−m

2T

k2 +m2(u†)m = (−1)k+1 (y0)k

k, ∀k ≥ 1.

In a compact form, it may be written as

QL∞u† = −(Y0)∞.

The matrix Q is the diagonal one defined in Remark 4.4 and L∞ is the infinite Loewner matrix

provided in Remark 5.1. The null-controllability problem has been studied in details in [19], where

the well posedness and the observability estimate are stated. This result is to relate to the invertibility

of the infinite Loewner matrix (see [7]).

6 Numerical Realizations

In order to check these theoretical predictions, we run first some computational investigations of the

singular values (σk)k∈N of the operator B. We examine the truncated matrix BN = (bkm)1≤k,m≤N

for large values of N , corresponding to the restriction of BB∗ to the N first Fourier modes. Their

eigenvalues are the squares of the singular values of the truncation of B. Calculations are realized

in Matlab. Next, we consider the controllability problem where the target yT is a parabolic pro-

file. The control u† is obtained by the inversion of the Cauchy system addressed in Section 5 after

regularization.

The first computations are accomplished when the cut-off N ranges from 5 to 24. The singular

values (σN )1≤k≤N are plotted in Figure 1 with different final times T = 10, 1, 0.1 and 0.01. Most

of them seem to spread towards zero when N grows higher. Although it is hardly perceptible

on these plots, the largest singular-values grow slowly towards infinity. In contrast, the smallest

ones decreases visually fast towards zero. Notice that when N exceeds 20, the lowest ones are

smaller than 10−8 and the accuracy suffers at that level. This might explain the clustering effect

observed on those singular values. Indeed, Matlab fails to approximate them and sends back

17

10−10

10−8

10−6

10−4

10−2

100

102

4

6

8

10

12

14

16

18

20

22

24

10−10

10−8

10−6

10−4

10−2

100

102

4

6

8

10

12

14

16

18

20

22

24

10−10

10−8

10−6

10−4

10−2

100

102

4

6

8

10

12

14

16

18

20

22

24

10−10

10−8

10−6

10−4

10−2

100

102

4

6

8

10

12

14

16

18

20

22

24

Figure 1: Singular values of BN for various cut-offs N . The up-left diagram is for T = 10, the up-right forT = 1, the down-left for T = 0.1 and the down-right for T = 0.01.

inaccurate values. To confirm these first trends, we depict in the right diagram of Figure 2, the

behavior of the largest singular value (σN )1 with respect to N for the two final times T = 1 and

T = 0.01. The shape of the curves and the (non-linear) regression suggests that (σN )1 behaves

rather like the upper bound predicted in (25) It grows up like√N . In the left panel we represent

in a linear-logarithmic scale, the singular values (σN )N , (σN )N−2 and (σN )N−4 where T = 1. The

curves are close to straight lines which suggests that those singular values decay exponentially fast

with respect to N . A careful look at the plots yields that for a fixed k, the sequences ((σN )N−k)N

increases like√N while the sequence ((σN )k)N decays exponentially fast toward zero for grow-

ing N . Another result confirmed by by numerical experimentation, we claim that an the fraction

of the singular values decays to zero is substantially more important than the fraction that grows

to infinity. Computations show that for kN = 0.1×N , we have that (σN )kN decreases toward zero.

This indicates that more than nine-tenth of the singular values are small. Another feature may be

noted from those plots. When the final time T is smaller the set of the singular values is noticeably

shifted towards zero. For instance, if T = 0.01, even for N = 5, the couple of smaller singular

values ((σ5)5, (σ5)5) are lower than 10−6. This is in agreement with the intuition one has on the

18

higher difficulty to reach the target yT in a shorter time T .

4 6 8 10 12 14 16 18 20

N

10-8

10-6

10-4

10-2

100

(σ ) _N _N

(σ ) _N _(N - 2)

(σ ) _N _(N - 4)

0 20 40 60 80 100

N

0

1

2

3

4

5

6

7

T = 1T = 0.01

Figure 2: The left panel depicts the lowest singular values ((σ2, (σN )N−4) of BN for various cut-offs N .Observe that the lowest (σN )N becomes erratic with Matlab at N = 19. The largest singular value (σN )1to the right.

We turn now to the effective solving of the controllability problem (28). Because of the exponen-

tial ill-conditioning of the truncated matrix, a crude inverting of it provides expectedly irrelevant

results. Indeed, any small (numerical) perturbations on the data yT or on the matrix entries may

have dramatic effects on the computations. Hence, to capture a reasonable control, a regularization

of the system is mandatory. We use the Tikhonov method by adding to the system a stabilizing

term so that we have to calculate the solution u† of the following regularized equation

(u†)k +∑

m≥1

1− e−(k2+m2)T

k2 +m2(u†)m = (−1)k+1π

2

(yT )kk

, ∀k ≥ 1. (31)

The real-number > 0 has to be selected suitably and automatically. Some popular rules can be

used such as the Discrepancy Principle, the General Cross Validation or the L-Curve (see [5, 29]).

This issue is familiar in ill-posed problems and still feeds a lot of scientific work. Addressing it with

the necessary care is beyond our scope. Only, notice that the ideal choice is the one that realizes

a balancing between the accuracy and the stabilization of the numerical computation.

The plots of Figure 3 correspond to the computations obtained by attempting to realize a parabolic

profile. The target is then fixed to

yT (x) = x(π − x), ∀x ∈ [0, π].

The panel to the le left depicts the approximated states realized by three numerical simulations

where the regularization parameter is equal to 10−2, 10−4 or 10−6. The second choice = 10−4

seems the best as it results on a profile that is close to the very one we want to reach, even at

the vicinity of x = π. The first = 10−2 results in an over-regularization where the shape of the

computed profile suffer form lesser accuracy. Even if the profile realized by under regularization

19

= 10−6 is closer to yT within a part of the domain I, it suffers clearly from an unacceptable

inaccuracy at the vicinity x = π. A look to the right panel where the controls are represented

confirms these observations. A under-regularization provokes useless energy expenses while an

over-regularization yields a decreasing in the amount of the energy to reach the target. To close,

we draw the attention to the fact that the smoothness of the control u† for t < T . The oscillations

around t = T suggest the presence of singularity at the final time. This is in accordance with

Remark 4.2.

0 0.5 1 1.5 2 2.5 3x

0

1

2

3

y(T

)

y_T

rho = 1e-02rho = 1e-04rho = 1e-06

0 0.2 0.4 0.6 0.8 1t

-20

0

20

40u(t

) rho = 1e-02rho = 1e-04rho = 1e-06

Figure 3: The left panel depicts the exact final state yT = x(π − x) and those computed using Tikhonovregularization with the parameter . The right panel draw the corresponding controls.

7 Conclusion

The issue of ill-posedness of the Dirichlet boundary controllability of the heat equation arises tech-

nical difficulties because the linear control operator has two properties that are somehow “opposite”

to each other. One is the unboundedness that is responsible of the fact that a fraction of its singular

values grows to infinity. The other is connected with the compactness and produces small singular

values. The spectrum of that operator is therefore spread from zero to infinity. Conducting Fourier

computations reduces the point to the study of some particularly structured matrices. The analysis

of their singular values concludes to the severe ill-posedness of the problem. In fact, things happens

as if the control operator is a direct sum of two sub-operators, one has a compact resolvent and the

other is compact with an exponential compactness degree that is the corresponding singular values

decrease exponentially fast towards zero.

20

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